\section{Recommendations}
\label{sec:recommendations}

The documentation proposes a staged plan to close the remaining gaps. Priorities are grouped below.

\subsection{Critical Fixes}
\begin{enumerate}
  \item Engage with the original authors to obtain clarifications on hidden-layer initialisation, normalisation, and any undocumented heuristics that may explain the accuracy gap.
  \item Implement the planned vectorised derivative computation for the 2D operator, reduce repeated forward passes, and profile the resulting speedup on smaller configurations before scaling up.
\end{enumerate}

\subsection{Validation Enhancements}
\begin{enumerate}
  \item Augment the solver with unit tests that validate operator application on analytical polynomials, enforce interface continuity, and check boundary condition residuals.
  \item Conduct convergence and sensitivity studies (e.g., subdomain refinement, collocation density, hidden-layer width) to characterise stability and conditioning.
\end{enumerate}

\subsection{Extensions and Optimisations}
\begin{enumerate}
  \item Add further PDE case studies, including nonlinear problems such as Burgers or Allen--Cahn equations, to exercise both solvers.
  \item Explore alternative autodiff backends (JAX) or sparse linear algebra integrations to improve scalability.
  \item Enhance visualisation and monitoring tooling to track residual norms, condition numbers, and runtime hotspots during training.
\end{enumerate}
